Factor completely.
$10 x^4+26 x^3-12 x^2$
We are given the polynomial \(10x^4 + 26x^3 - 12x^2\) and our objective is to factor it completely. A "complete" factorization means expressing the polynomial as a product of simpler, irreducible polynomials over the rational numbers. We begin by examining the terms to identify any common factors.
First, we identify the greatest common factor (GCF) of the terms \(10x^4\), \(26x^3\), and \(-12x^2\). All coefficients (10, 26, -12) are divisible by 2, and every term contains at least \(x^2\). Thus, the GCF is \(2x^2\). Factoring out \(2x^2\), we obtain:
\[10x^4 + 26x^3 - 12x^2 = 2x^2(5x^2 + 13x - 6).\]
Next, we focus on factoring the quadratic trinomial \(5x^2 + 13x - 6\). We look for two numbers that multiply to \(ac = 5 \times (-6) = -30\) and add to \(b = 13\). By examining pairs of factors of \(-30\), we find that \(15\) and \(-2\) satisfy both conditions: \(15 \times (-2) = -30\) and \(15 + (-2) = 13\). We can now split the middle term:
\[5x^2 + 15x - 2x - 6.\]
Grouping the terms, we get:
\[(5x^2 + 15x) - (2x + 6) = 5x(x + 3) - 2(x + 3).\]
Factoring out the common binomial \((x + 3)\), we obtain:
\[(5x - 2)(x + 3).\]
Finally, we combine the GCF factored out in step 2 with the binomials obtained in step 3 to write the completely factored form of the original polynomial:
\[10x^4 + 26x^3 - 12x^2 = 2x^2(5x - 2)(x + 3).\]
This expression is now factored into its simplest irreducible components over the rational numbers. Therefore, the completely factored form is \(\boxed{2x^2(5x - 2)(x + 3)}\).
Factoring polynomials is a fundamental tool in algebra used to simplify complex equations, solve for roots, and analyze function behavior. For example, if you are designing a storage box where the volume is given by a polynomial, factoring that volume expression can help you determine the physical dimensions (length, width, and height) in terms of a variable \(x\). This makes it easier to optimize the shape or calculate how much material is needed for production.